Achievements
Complete each achievement to receive the allotted gamerscore:

The Soldier We Need You To Be (25) - Completed the Campaign on Normal difficulty.
Folks Need Heroes... (50) - Completed the Campaign on Heroic difficulty.
Gods Must Be Strong (125) - Completed the Campaign on Legendary difficulty.
This Is Not Your Grave (10) - Completed the 9th mission on Normal or harder.
Send Me Out... With A Bang (10) - Completed the 10th mission on Normal or harder.
They've Always Been Faster (25) - Cleared the 2nd mission without setting foot in a drivable vehicle.
Two Corpses In One Grave (25) - Killed 2 vehicles at once with the Target Locator in the 3rd mission.
Banshees, Fast And Low (25) - Hijacked a Banshee during the Reach Campaign.
Your Heresy Will Stay Your Feet (25) - Killed the Elite Zealot before he escaped during the 5th mission.
If They Came To Hear Me Beg (25) - Performed an Assassination against an Elite to survive a fall that would've been fatal.
Wake Up Buttercup (25) - Destroyed the Corvette's engines & escort in under 3 minutes in the 6th mission on Heroic or harder.
Tank Beats Everything (25) - Finished the 9th mission on Legendary with the Scorpion intact.
Lucky Me (25) - Earned a Triple Kill while Jetpacking in Campaign, Firefight or Matchmaking.
A Monument To All Your Sins (150) - Completed every mission in Halo:: Reach alone, on Legendary.
We're Just Getting Started (10) - Completed the 2nd mission on Normal or harder.
Protocol Dictates Action (10) - Completed the 3rd mission on Normal or harder.
I Need A Weapon (10) - Completed the 4th mission on Normal or harder.
To War (10) - Completed the 5th mission on Normal or harder.
You Flew Pretty Good (10) - Completed the 6th mission on Normal or harder.
Into The Howling Dark (10) - Completed the 7th mission on Normal or harder.
Dust And Echoes (10) - Completed the 8th mission on Normal or harder.
KEEP IT CLEAN (5) - Killed 7 Moa during the 2nd mission of the Campaign.
I Didn't Train To Be A Pilot (10) - Killed 3 of the anti-aircraft batteries during the 8th mission.
Crowd Control (10) - Earned a Killionaire medal in Firefight.
Knife To A Gun Fight (5) - As an Elite, killed 5 Spartan players in Matchmaking.
Score Attack (10) - Scored 15,000 points in Score Attack Firefight Matchmaking.
Firestarter (10) - Scored 50,000 points in a Firefight game.
Doctor, Doctor (5) - Used a Health Pack to replenish life after taking body damage.
That's A Knife (10) - Performed an Assassination on an enemy.
I See You Favour A .45 (10) - Killed 10 enemies in a Firefight or Campaign session with the M6G pistol.
An Elegant Weapon (10) - Killed 10 enemies in a Firefight or Campaign session with the DMR.
Swap Meet (10) - Traded weapons with an AI ally in Campaign.
A Spoonful Of Blamite (10) - Killed 10 enemies in Firefight or Campaign with a supercombine explosion.
Be My Wingman Anytime (5) - Let a teammate spawn on you 5 times in an Invasion Matchmaking game.
Yes, Sensei (10) - Earned a First Strike Medal in a Matchmaking game.
Skunked (10) - Won a game of Invasion in the 1st phase.
What's A Killing Spree? (5) - Earned a Killing Spree in multiplayer Matchmaking.
Blaze Of Glory (25) - Scored 200,000 points in a Firefight game.
Heat In The Pipe (75) - Scored 1,000,000 points in a Firefight game.
Game, Set, Match (25) - Completed a Firefight set on Legendary without dying.
Make It Rain (10) - Purchased an item from the Armory that required the rank of Lt. Colonel.
The Start Of Something (15) - Reached the rank of Corporal in the UNSC.
An Honor Serving (25) - Reached the rank of Captain in the UNSC.
A Storage Solution (5) - Used the File Browser to upload a file to your File Share.
A New Challenger (10) - Completed all of the Daily Challenges in a given day.
Make It Drizzle (10) - Purchased an item from the Armory.
Cool File, Bro (5) - Recommended a file to someone.
Lemme Upgrade Ya (10) - Advanced a Commendation to a Silver state.
One Down, 51 To Go (10) - Completed a Weekly Challenge.

Additionally, there are seven Noble Map Pack DLC achievements:

Both Barrels (50) - Earn a Double Kill with the shotgun in multiplayer Matchmaking.
Poppin' & Lockin' (25) - Destroy a vehicle using Armor Lock in a matchmade game.
Totally Worth It (50) - Earn a Double Kill from the Grave in multiplayer Matchmaking.
Cross-Mappin' (50) - Kill a player at long range with the DMR in a matchmade Slayer game.
Offensive Driver (20) - Earn a kill in a matchmade Rocket Race game.
You Blew It Up! (13) - Blow up the research facility in a matchmade Invasion game on Breakpoint.
You Ate All the Chips (42) - Collect all of the flags in a matchmade Stockpile game.

Additionally, there are three Defiant Map Pack DLC achievements:

Crackin' Skulls (45) - Kill 3 players with the Oddball in a matchmade Oddball game.
Return to Sender (30) - Successfully return two flags in a matchmade CTF game.
Candy from a Baby (75) - Capture two flags from an enemy capture zone in a matchmade Stockpile game.

Unlock Avatar Awards
Carter's Helmet - Clear a Campaign mission on Legendary without dying.
Emile's Helmet - Earned a Bulltrue medal in either multiplayer or Firefight Matchmaking.
Jorge's Helmet - Earn a Killtacular in multiplayer Matchmaking.
Jun's Helmet - Kill 100 enemies in a row without dying in either the Campaign or Firefight.
Kat's Helmet - Avenged a teammate's death in multiplayer Matchmaking.

Unlock Carter's Helmet Easily
Normally, you'd have to complete a level on Legendary without dying to get this. However, you can play up until just before the end and "Save and Quit" on a checkpoint. Then, Resume the game and finish the level (without dying for those last few parts). After you finish, you should get the Avatar Unlock pop-up.